For geometric series $(a_n)$ with all items real, if $S_{10}=10,S_{30}=70$, then $S_{40}=$ $\text{(A)}150\qquad\text{(B)}-200\qquad\text{(C)}150\text{ or }-200\qquad\text{(D)}-50\text{ or }400$ Note: $S_n=\sum_{i=1}^{n}a_i$.

$ 20 $ discs of radius $ 1 $ are bounded by a circle of radius $ 10. $ Show that in the interior of this circle is sufficient space to insert $ 7 $ discs of radius $ \frac{1}{3} $ that doesn't touch any other disc. [i]Flavian Georgescu[/i]

Find all possible pairs of real numbers $ (x, y) $ that satisfy the equalities $ y ^ 2- [x] ^ 2 = 2001 $ and $ x ^ 2 + [y] ^ 2 = 2001 $.

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$. It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$. Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$.

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Let $ABCD$ be a convex quadrilateral. The common external tangents to circles $(ABC)$ and $(ACD)$ meet at point $E$, the common external tangents to circles $(ABD)$ and $(BCD)$ meet at point $F$. Let $F$ lie on $AC$, prove that $E$ lies on $BD$.

Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

Alice thinks about a natural number in her mind. Bob tries to find that number by asking him the following 10 questions: [list] [*]Is it divisible by 1? [*]Is it divisible by 2? [*]Is it divisible by 3? [*]... [*]Is it divisible by 9? [*]Is it divisible by 10? [/list] Alice's answer to all questions except one was "yes". When she answers "no", she adds that "the greatest common factor of the number I have in mind and the divisor in the question you asked is 1”. According to this information, to which question did Alice answer "no"?

Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$. [i](tatari/nightmare)[/i]

A sequence $(a_n)$ is defined by $a_0=-1,a_1=0$, and $a_{n+1}=a_n^2-(n+1)^2a_{n-1}-1$ for all positive integers $n$. Find $a_{100}$.

\[\int_{-\infty}^{0} \frac{1}{e^{-x}+2e^{x}+e^{3x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$. (1) Find $a_n$. (2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$. Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$. Last Edited

Let $ABC$ be an acute triangle. Construct a point $X$ on the different side of $C$ with respect to the line $AB$ and construct a point $Y$ on the different side of $B$ with respect to the line $AC$ such that $BX=AC$, $CY=AB$, and $AX=AY$. Let $A'$ be the reflection of $A$ across the perpendicular bisector of $BC$. Suppose that $X$ and $Y$ lie on different sides of the line $AA'$, prove that points $A$, $A'$, $X$, and $Y$ lie on a circle.

Let $a_1, a_2, ..., a_n$ be real numbers such that $a_1 = a_n = a$ and $a_{k+1} \le \frac{a_k + a_{k+2}}{2} $, for all $k = 1, 2, ..., n - 2$. Prove that $a_k \le a,$ for all $k = 1, 2, ..., n.$

Suppose there are $n$ points on the plane, no three of which are collinear. Draw $n-1$ non-intersecting segments (except possibly at endpoints) between pairs of points, such that it is possible to travel between any two points by travelling along the segments. Such a configuration of points and segments is called a [i]network[/i]. Given a network, we may assign labels from $1$ to $n-1$ to each segment such that each segment gets a different label. Define a [i]spin[/i] as the following operation: $\bullet$ Choose a point $v$ and rotate the labels of its adjacent segments clockwise. Formally, let $e_1,e_2,\cdots,e_k$ be the segments which contain $v$ as an endpoint, sorted in clockwise order (it does not matter which segment we choose as $e_1$). Then, the label of $e_{i+1}$ is replaced with the label of $e_{i}$ simultaneously for all $1 \le i \le k$. (where $e_{k+1}=e_{1}$) A network is [i]nontrivial[/i] if there exists at least $2$ points with at least $2$ adjacent segments each. A network is [i]versatile[/i] if any labeling of its segments can be obtained from any initial labeling using a finite amount of spins. Find all integers $n \ge 5$ such that any nontrivial network with $n$ points is versatile. [i]Proposed by Yeoh Zi Song[/i]

Find the largest possible value of $a$ such that there exist real numbers $b,c>1$ such that \[a^{\log_b c}\cdot b^{\log_c a}=2023.\] [i]Proposed by Howard Halim[/i]

Source: 2017 Canadian Open Math Challenge, Problem B4 ----- Numbers $a$, $b$ and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.

We say that a prime $p$ is $\textit{philé}$ if there is a polynomial $P$ of non-negative integer coefficients smaller than $p$ and with degree $3$, that is, $P(x) = ax^3 + bx^2 + cx + d$ where $a, b, c, d < p$, such that $$\{P(n) | 1 \leq n \leq p\}$$ is a complete residue system modulo $p$. Find all $\textit{philé}$ primes. Note: A set $A$ is a complete residue system modulo $p$ if for every integer $k$, with $0 \leq k \leq p - 1$, there exists an element $a \in A$ such that $$p | a-k.$$

Which of the following is closest to $\sqrt{65}-\sqrt{63}$? $ \textbf{(A)}\ .12 \qquad\textbf{(B)}\ .13 \qquad\textbf{(C)}\ .14 \qquad\textbf{(D)}\ .15 \qquad\textbf{(E)}\ .16 $

You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.

A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year $ 2003$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

There are 20 blue points on the circle and some red inside so no three are collinear. It turned out that there exists $1123$ triangles with blue vertices having 10 red points inside. Prove that all triangles have 10 red points inside

For each positive integer $n$, let $rad(n)$ denote the product of the distinct prime factors of $n$. Show that there exists integers $a,b > 1$ such that $gcd(a,b)=1$ and $$rad(ab(a+b)) < \frac{a+b}{2024^{2024}}$$. For example, $rad(20)=rad(2^2\cdot 5)=2\cdot 5=10$.

Compute all real solutions $(x,y)$ with $x\geq y$ that satisfy the pair of equations \begin{align*} xy&=5\\ \frac{x^2+y^2}{x+y}&=3. \end{align*}